Applications of degree estimate for subalgebras

Abstract

Let $K$ be a field of positive characteristic and $K$ be the free algebra of rank two over $K$. Based on the degree estimate done by Y.-C. Li and J.-T. Yu, we extend the results of S.J. Gong and J.T. Yu's results: (1) An element $p(x,y)\in K$ is a test element if and only if $p(x,y)$ does not belong to any proper retract of $K$; (2) Every endomorphism preserving the automorphic orbit of a nonconstant element of $K$ is an automorphism; (3) If there exists some injective endomorphism $\phi$ of $K$ such that $\phi(p(x,y))=x$ where $p(x,y)\in K$, then $p(x,y)$ is a coordinate. And we reprove that all the automorphisms of $K$ are tame. Moreover, we also give counterexamples for two conjectures established by Leonid Makar-Limanov, V. Drensky and J.-T. Yu in the positive characteristic case.Comment: 12 page